Uniform Convergence Rates for Maximum Likelihood Estimation under Two-Component Gaussian Mixture Models

TL;DR

This paper establishes uniform convergence rates and minimax lower bounds for maximum likelihood estimation in two-component Gaussian mixture models without assuming component separation, revealing a phase transition based on mixture balance.

Contribution

It provides the first uniform convergence rates for MLE in two-component Gaussian mixtures with unequal variances, highlighting the impact of mixture balance on estimation rates.

Findings

Convergence rates depend on mixture balance.

Phase transition in estimation accuracy.

Simulation confirms theoretical results.

Abstract

We derive uniform convergence rates for the maximum likelihood estimator and minimax lower bounds for parameter estimation in two-component location-scale Gaussian mixture models with unequal variances. We assume the mixing proportions of the mixture are known and fixed, but make no separation assumption on the underlying mixture components. A phase transition is shown to exist in the optimal parameter estimation rate, depending on whether or not the mixture is balanced. Key to our analysis is a careful study of the dependence between the parameters of location-scale Gaussian mixture models, as captured through systems of polynomial equalities and inequalities whose solution set drives the rates we obtain. A simulation study illustrates the theoretical findings of this work.